Question

Ap calculus ab. I'm completely lost

Answer 1

Answer:

$\displaystyle f'(0) = \frac{4}{3}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Terms/Coefficients
• Factoring
• Functions
• Function Notation
• Exponential Rule [Rewrite]:                                                                              $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]:                                                                  $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                            $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$  

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                    $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Explanation:

Step 1: Define

Identify

$\displaystyle f(x) = (x^2 - 2x - 1)^\bigg{\frac{2}{3}}$

Step 2: Differentiate

1. Chain Rule:                                                                                                        $\displaystyle f'(x) = \frac{d}{dx}[(x^2 - 2x - 1)^\bigg{\frac{2}{3}}] \cdot \frac{d}{dx}[(x^2 - 2x - 1)]$
2. Basic Power Rule {Derivative Property - Subtraction]:                                  $\displaystyle f'(x) = \frac{2}{3}(x^2 - 2x - 1)^\bigg{\frac{2}{3} - 1} \cdot (2x^{2 - 1} - 2x^{1 - 1} - 0)$
3. Simplify:                                                                                                             $\displaystyle f'(x) = \frac{2}{3}(x^2 - 2x - 1)^\bigg{\frac{-1}{3}} \cdot (2x - 2)$
4. Rewrite [Exponential Rule - Rewrite]:                                                              $\displaystyle f'(x) = \frac{2(2x - 2)}{3(x^2 - 2x - 1)^\bigg{\frac{1}{3}}}$
5. Factor:                                                                                                               $\displaystyle f'(x) = \frac{4(x - 1)}{3(x^2 - 2x - 1)^\bigg{\frac{1}{3}}}$

Step 3: Evaluate

1. Substitute in x [Derivative]:                                                                              $\displaystyle f'(0) = \frac{4(0 - 1)}{3[0^2 - 2(0) - 1]^\bigg{\frac{1}{3}}}$
2. [Order of Operations] Simplify:                                                                       $\displaystyle f'(0) = \frac{4}{3}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e